I really don't know how to do this. Usually you find n basis vectors of X => dim(X) = n. But I find it pretty hard, anyone got some ideas? Any help would be much appreciated.

The space F is not finite-dimensional. In fact, let . Then G is infinite-dimensional. (For example, it will contain all the functions in that have compact support.) But F is the image of G under the inverse Fourier transform, so F is also infinite-dimensional.